# Student Projects

## Consequences of Additional Parameters within Sudoku Boards

#### Faculty Mentor: Dr. Nathan Shank

For this project we explore the process of solving Sudoku boards with various parameters in a simpler manner. Our group solves multiple example boards with established restrictions, analyzes patterns within individual cells as well as on a larger scale to establish a basis for our research. By isolating certain restrictions, we created algorithms which solve the boards satisfying the initial conditions set forward by the parameters established. When combining these algorithms, a set of conditional statements can be set forward, wherein a wider variety of possible boards can be solved with greater ease and efficiency.

## Fixed Cram with Higher Polyominoes (Cramominoes)

#### Faculty Mentor: Dr. Nathan Shank

A polyomino is a set of n cells such as a domino or a tetromino from the popular game Tetris. Cram is a well-known impartial combinatorial game where each player places a domino, orientated either horizontally or vertically, on a square grid. We define Fixed Cram as only being played with horizontal or vertical pieces such that the rotated version of the piece is a different piece. We explore optimal strategies for playing Fixed Cram with higher polyominoes. In the game play, we set the number of players, board size, and singular game piece prior to the start of the game. In order to win our game, you need to be the last person to lay a polyomino so that the next player cannot lay another piece. In this project, we examine the patterns between symmetrical and asymmetrical pieces on specific square board sizes.

## Introduction to Computational Biology

#### Faculty Mentor: Dr. Nathan Shank

This project served as an introduction to computational biology from the mathematical perspective. A short introduction to biology was followed by a rigorous discussion of gene mapping and algorithms. Other areas discussed included sequencing related to DNA, RNA and generalizing these to other mathematical models. We also spent time looking at probabilistic aspects of sequence alignment.

## Chromatic Reduction

#### Faculty Mentor: Dr. Nathan Shank

Throughout this project we will be looking at different kinds of simple undirected graphs. We discuss how to reduce the chromatic number of a graph and how to reduce the chromatic index of a graph by removing edges or ver- tices. We look at specific graph classes and find the minimum number of edges/vertices needed to be removed to reduce the chromatic number/index by some positive integer k. We also provide examples of graphs where reducing the parameter stepwise is not always the optimal strategy.

## Strategies for Dots-and-Lines Using Nontraditional Rules

#### Faculty Mentor: Dr. Nathan Shank

For decades, the simple game of Dots-and-Lines has been played by children and adults alike, establishing it as one of the most popular paper- and-pencil games in the world. In the traditional version of the game, when a player closes a box, that player then makes another move anywhere on the board. For our research, we wanted to investigate the effects of the players not being able to make another move after completing a box. Simplifying the rules in this sense allowed us to deduce which player would always win depending on the dimensions of the board. We further developed our findings by modifying the rules of the game to allow players to move on the diagonals of rectangular boards, as well as by applying modifications of the rules of the game to games which use triangular boards.

## Nonograms

#### Faculty Mentor: Dr. Nathan Shank

Nonograms is a classic logic puzzle in which the squares of a m×n grid are to be filled in a determined way to display a picture. What determines which squares to color is sequences of numbers above each column and to the left of each row. The numbers describe how many consecutive squares are to be colored in that row or column, and multiple numbers represent multiple blocks of colored squares, with at least one uncolored square in between blocks. Our project studies and counts boards that, given these instructions, can get colored in more than one way.

## Clumsy Packing with Polyominoes

#### Faculty Mentor: Dr. Nathan Shank

For this summer SOAR Project, Emma worked as an undergraduate researcher for a REU program at Moravian University. Her work in this project related to packing polyominoes into a fixed square in the least efficient way possible. The group analyzed T, L, and rectangle polyominoes.

## Almost Dominoes

#### Faculty Mentor: Dr. Nathan Shank

For this project, we investigate how to finish a game of dominoes using the least amount of pieces given two parameters p and q. The parameter q represents the number of dominoes with the same pattern, and parameter p represents the number of maximum dots/pips possible on one side of a piece (Note each piece has two sides to it). With these two parameters, we find a formula that tells us the minimum number of dominoes that are needed to finish the game.

## Nim on Graphs

#### Faculty Mentor: Dr. Nathan Shank

In this project, we explore optimal strategies for playing the game of Nim on path graphs. In Nim, players take turns picking up chips from various piles. By extending this to path graphs, we create the restriction that the next player can only pick up chips from a pile adjacent to the previous player’s pile. We focused on path graphs of length 3 and 4 to help us identify patterns in larger size graphs.

## Quoridor

#### Faculty Mentor: Dr. Nathan Shank

Quoridor is a 2-4 player strategy board game. In the game, players take turns either moving their pawn a single space or placing a fence, with the goal of getting their pawn to the opposing side first. In the hopes of determining an optimal strategy for the game, we limited board size and the number of fences for each player. Through looking at 3x3 and 4x4 size boards, we developed a few theories that helped us explore larger board sizes.

## Polyominoes

#### Faculty Mentor: Dr. Nathan Shank

A polyomino is a connected plane geometric figure made with squares so that two squares must intersect along an entire edge. For example, the pieces in the game tetris. In this project, we explore the possible configurations of polyominoes on various board sizes. We analyzed one type of polyomino, consisting of three or four cells to fill different size boards. We also used multiple combinations of different types of polyominoes, but only with square boards. We researched on the n=3 and n=4, using multiple configurations with the goal of filling a board. We also have research targeting certain types of pieces (such as the "S-Piece") regarding how much one of those pieces can fill a board.

## Increasing the Independence Number

#### Faculty Mentor: Dr. Nathan Shank

We analyzed the number of edges required to be added to or removed from a simple graph in order to decrease or increase the vertex independence number of the graph.

## Changing the Chromatic Number and Index of a Graph

#### Faculty Mentor: Dr. Nathan Shank

The chromatic number of a graph is the least number of colors needed to color the vertices of a graph so that no two adjacent vertices have the same color. The index is similarly defined for edges. In this project we were finding the minimum number of vertices (or edges) that needed to be removed in order to increase the chromatic number (or index) by some fixed value.

## Differential Equations on Path Algebras

#### Faculty Mentor: Dr. Shannon Talbott

Just as we can take a derivative using the product rule in Calculus, we can extend this idea beyond the derivative of a product of functions and can define a derivation on an algebra. This area of mathematics, called differential algebra, was introduced by J.F. Ritt in the 1950s. However, much of the work has been focused in the commutative setting (meaning order of multiplication does not matter). The focus of this Honors project was to consider path algebras, which is in the non-commutative setting, and solve a second-order differential equation.

## Counting Planar Graphs

#### Faculty Mentor: Dr. Nathan Shank

For this project, we expanded our research capabilities by conducting research on graph theory. We analyzed the total number of unlabeled, planar graphs for specific degrees x or y. In terms of our individual research, our primary focus was when x = 1 and y = 2. Our goal was to find the total number of graphs with the given parameters x and y. We constructed an algorithm for any graph of order n, while n is the number of nodes of the graph. With this information, we then analyzed patterns as the value of n increased and came up with a conjecture to find the estimated value for larger sized graphs.

## Magic Square of Squares

#### Faculty Mentor: Dr. Nathan Shank

Our project focuses on magic squares. These are grids, usually a 3×3, filled in with integers such that each row, column, and diagonal adds up to the same number. However, no one has found a 3×3 magic square where each element is a perfect square, which is where an integer’s square root is also an integer. We are set out to find a solution, if it exists.

## Games on Graphs

#### Faculty Mentor: Dr. Nathan Shank

For this project, we are focusing on edge deletion games on path graphs. With three players, each player takes a turn by removing an edge from a graph. The player that is forced to delete an edge and leaves a vertex isolated will lose the game. If you don’t see a way to win the game for yourself, you try your best to make sure that you don’t lose.

## Squares of Graphs

#### Faculty Mentor: Dr. Nathan Shank

In this project, we are exploring a topic of graph theory that focuses on the square of a directed graph. A standing conjecture states that for every oriented graph there is a vertex whose out-degree at least doubles in the square of the graph. We explore the different qualities of a square graph, while we work our way towards providing evidence the conjecture is indeed true.

## Reducing Cardinality Redundancy In P(n,2) Peterson Graphs

#### Faculty Mentor: Dr. Nathan Shank

The dominating number of a graph is the minimum number of vertices needed so that every vertex in the graph is either in the dominating set or is adjacent to a vertex in the dominating set. In many graphs this means that there are vertices which are dominated more than once and/or the total number of vertices adjacent to the dominating set is very large, meaning some vertices may be highly over-dominated. In this project we try to find the ways to efficiently dominate Peterson graphs. If they can not be efficiently dominated, we try to find the dominating set which is the closest to efficient (Cardinality Redundancy).

## Determining College Success Based on High School Performance: Fitting Linear Mixed Models

#### Faculty Mentor: Dr. Brenna Curley

One of the challenges that college Admissions Offices face revolves around ensuring that retention rates rise or at least remain the same. In this Honors project, we attempt to predict student success at Moravian University. We use linear mixed effects models to describe the effects of high school Grade Point Average (GPA), SAT scores, and gender on college GPA; we additionally account for potential differences between high schools.

## Edge and Vertex Failure Games on Graphs

#### Faculty Mentor: Dr. Nathan Shank

This Honors project looked at problems that arise from the synthesis of Game Theory and Graph Theory. We analyzed a partizan combinatorial game where one player moves by deleting a single vertex and the other player moves by deleting a single edge from a graph. We then establish variants of this game by altering the sets of graphs that are unplayable for each player. We were able to solve this game and its variants on several graph classes including paths, wheels, and complete graphs.

## Partizan Graph Connectivity Games

#### Faculty Mentor: Dr. Nathan Shank

For this Summer SOAR Project, Devon worked as an undergraduate researcher for a REU hosted by Muhlenberg College. The REU was focused on working with the Online Encyclopedia of Integer Sequences (OEIS). He worked as part of a Graph Theory research team that examined sequences that arose from the Edge Distinguishing Chromatic Number of different graph classes. He also worked as part of a Game Theory research team that expanded upon research done in component deletion games played on graphs.

## Explorations of Game Theory and Properties of Factor Pair Latin Squares

#### Faculty Mentor: Dr. Nathan Shank

This Summer SOAR research project was conducted at the 2017 Muhlenberg College Mathematics and Computer Science Summer Research Experience for Undergraduates Program (REU). Makkah worked on two unrelated research projects, one exploring properties of Factor Pair Latin Squares and one exploring Game Theory. The game theory project, in which we created a partisan variant of the popular combinatorial game, Nim, is played on a colored cycle graph. The bulk of our research was devoted to analyzing and solving our game.

## Subtractions Games

#### Faculty Mentor: Dr. Michael Fraboni

This semester SOAR project investigated questions in Combinatorial Game Theory. In particular, we determined the value of game positions for a family of subtraction games based on Beatty sequences.

## Clever Problem Solving II

#### Faculty Mentor: Dr. Nathan Shank

This project develops additional problem solving skills by developing a series of 14 lessons based on topics seen in mathematical problem solving. We learn creative ways to approach interesting mathematical problems. Topics include: Generating Functions, Recurrence Relations, Probability Problems, Weird Calculus, Diophantine Equations, Games, Combinatorics, Hyperoperations, Permutations, and Divisibility Tricks

## Probability Theory

#### Faculty Mentor: Dr. Nathan Shank

This project was centered around learning the fundamentals of Probability Theory which is at the intersection of probability and analysis. Topics covered included Random Variables, Weak and Strong Law of Large Numbers, Properties of Convergence, Generating Functions, Markov Chains, and Random Walks.

## Data and Soul: Introductory Data Mining and Data Analysis on Music Data

#### Faculty Mentor: Dr. Thyago Mota, Dr. Brenna Curley

In this work, presented at the 2018 National Conference on Undergraduate Research (NCUR), we investigate how songs lyrically and musically influence popularity and emotional responses throughout contemporary history. We build a dataset of 27,346 songs that are listed on the Billboard “Hot 100” list from 1958 to 2017. We then use Spotify’s song metrics, together with a weighted sampling function, to evaluate how music changed over time.

## Exploration of Potential Energy Functions of the 4 3𝝥0 Electronic State of NaCs

#### Faculty Mentor: Dr. Shannon Talbott, Dr. Ruth Malenda

In Physics, one area of research is determining the electronic potential energy curves of diatomic molecules. In this Honors project focused on the diatomic molecule sodium-cesium (NaCs), namely the 4 3𝝥0 electronic state of NaCs, which has a complicated potential structure with a double well. Using experimental data from collaborators, functional forms of the double well energy functions were explored using two approaches. Initially, the double well electronic state was considered as a combination of simple, standard single well forms with a switching term to transition between single wells. Then, a Spline Exponential-Morse Long Range potential function was found using a program called betaFIT.

## An Exploration of Lie Algebras and Kostant’s Weight Multiplicity Formula

#### Faculty Mentor: Dr. Shannon Talbott

Lie algebras are particular vector spaces with applications to physics like representing symmetries in atoms such as hydrogen as well as applications to special and general relativity. Just as eigenvectors and eigenspaces can be used to decompose a vector space into subspaces, weights and weight spaces can be used to decompose a Lie algebra into subalgebras, in certain cases. This decomposition can help us to understand these particular Lie algebras. Kostant’s multiplicity formula is concerned with weights, and while it appears to be simplistic, calculations are often extremely difficult. The aim of this Honors project was to add to the understanding of the multiplicity formula.

## Quasi-Crowns

#### Faculty Mentor: Dr. Shannon Talbott

In graph theory, one problem that has motivated research for well over a century is the graph coloring problem. That is, what is the minimum number of colors required to color the vertices of a graph so that no adjacent vertices are the same color? This problem is an NP-complete problem, but there are particular families of graphs where this question has be answered. One such family of graphs comes from a particular type of partially ordered set called a crown. This SOAR project explored the notion of layering such crowns.

## Nutrient Intake of Dancers: A Measurement Error Analysis

#### Faculty Mentor: Dr. Brenna Curley

Diet plays an important role in any athlete’s overall health. Many studies seek to understand the relationship between an athlete’s daily intake and performance quality; however, these studies are not frequently conducted for performance-based activities such as dance. Further, existing studies relating dancers' nutrient intake and health fail to account for the error in measuring long-term average intake. In this Honors project, we propose using a measurement error model to account for this error. As an application, we analyze the relationship between dietary intake, body composition, and energy levels of pre-professional contemporary dancers.

## Similarity Analysis of Hymns using Polynomial Regressions

#### Faculty Mentor: Dr. Nathan Shank

In this honors project, we developed software which is used to fit polynomials to music in order to analyze similarities and differences between composers. We used the software to look at mathematical differences of these polynomials and then interpreted these findings in a musical context.